Differential operators and Higher Specht polynomials

In this note, we study the action of the rational quantum Calogero-Moser system on polynomials rings. This a continuation of our paper [Ibrahim Nonkan 2021 J. Phys.: Conf. Ser. 1730 012129] in which we deal with the polynomial representation of the ring of invariant differential operators. Using the higher Specht polynomials we give a detailed description of the actions of the Weyl algebra associated with the ring of the symmetric polynomial C[x 1,..., xn]Sn on the polynomial ring C[x 1,..., xn ]. In fact, we show that its irreducible submodules over the ring of differential operators invariant under the symmetric group are its submodules generated by higher Specht polynomials over the ring of the symmetric polynomials. We end up studying the polynomial representation of the ring of differential operators invariant under the actions of products of symmetric groups by giving the generators of its simple components, thus we give a differential structure to the higher Specht polynomials.

Differential operators and reection group of type Dn

In this note, we study the actions of rational quantum Olshanetsky-Perelomov systems for finite reflections groups of type D n . we endowed the polynomial ring C[x 1,..., xn ] with a differential structure by using directly the action of the Weyl algebra associated with the ring C[x 1,..., xn ] W of invariant polynomials under the reflections groups W after a localization. Then we study the polynomials representation of the ring of invariant differential operators under the reflections groups. We use the higher Specht polynomials associated with the representation of the reflections group W to exhibit the generators of its simple components.

Differential operators and reection group of type Bn

In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system for a finite reflection group W of type Bn. We endowed the polynomial ring C[x 1,..., xn ] with a structure of module over the Weyl algebra associated with the ring C[x 1,..., xn]W of invariant polynomials under a reflections group W of type Bn . Then we study the polynomials representation of the ring of invariant differential operators under the reflections group W. We make use of the theory of representation of groups namely the higher Specht polynomials associated with the reflection group W to yield a decomposition of that structure by providing explicitly the generators of its simple components.

Actions of the additive group Ga on certain noncommutative deformations of the plane

We connect the theorems of Rentschler [18] and Dixmier [10] on locally nilpotent derivations and automorphisms of the polynomial ring A 0 and of the Weyl algebra A 1, both over a field of characteristic zero, by establishing the same type of results for the family of algebras A h = 〈 x , y | y x − x y = h ( x ) 〉 , {A_h} = \left\langle {x,y|yx - xy = h\left( x \right)} \right\rangle , , where h is an arbitrary polynomial in x. In the second part of the paper we consider a field 𝔽 of prime characteristic and study 𝔽[t]-comodule algebra structures on Ah . We also compute the Makar-Limanov invariant of absolute constants of Ah over a field of arbitrary characteristic and show how this subalgebra determines the automorphism group of Ah .

Centralizers of Rank One in the First Weyl Algebra

Centralizers of rank one in the first Weyl algebra have genus zero.

Sharing symmetries in non-linear systems: Generalized Heisenberg–Weyl algebra on the de Sitter space-time out of the sphere S3

In this paper, we exploit the formal equivalence of the Solution Manifold of two distinct physical systems to create enough symmetries so as to characterize them by Noether Invariants, thus favoring their future quantization. In so doing, we somehow generalize the Arnold Transformation for non-necessarily linear systems. Very particularly, this algorithm applies to the case of the motion on the de Sitter space-time providing a finite-dimensional algebra generalizing the Heisenberg–Weyl algebra globally. In this case, the basic (contact) symmetry is imported from the motion of a (non-relativistic) particle on the sphere [Formula: see text].